### Embedding C(X) in an algebra of analytic functions

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We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra ${L}^{1}\left(G\right)$ of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients $A\left(\Gamma \right)/\overline{J\left(E\right)}$ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct...

We modify the very well known theory of normed spaces (E,||·||) within functional analysis by considering a sequence (||·||ₙ: n ∈ ℕ) of norms, where ||·||ₙ is defined on the product space Eⁿ for each n ∈ ℕ. Our theory is analogous to, but distinct from, an existing theory of ’operator spaces’; it is designed to relate to general spaces ${L}^{p}$ for p ∈ [1,∞], and in particular to L¹-spaces, rather than to L²-spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach...

We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space ${A}^{\left(n\right)}$ is zero; i.e., ${\mathscr{H}}^{1}(A,{A}^{\left(n\right)})=0$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study...

In this memoir, we shall consider weighted convolution algebras on discrete groups and semigroups, concentrating on the group (ℚ,+) of rational numbers, the semigroup $({\mathbb{Q}}^{+\u2022},+)$ of strictly positive rational numbers, and analogous semigroups in the real line ℝ. In particular, we shall discuss when these algebras are Arens regular, when they are strongly Arens irregular, and when they are neither, giving a variety of examples. We introduce the notion of ’weakly diagonally bounded’ weights, weakening the known...

In recent years, there have been several studies of various ’approximate’ versions of the key notion of amenability, which is defined for all Banach algebras; these studies began with work of Ghahramani and Loy in 2004. The present memoir continues such work: we shall define various notions of approximate amenability, and we shall discuss and extend the known background, which considers the relationships between different versions of approximate amenability. There are a number of open questions...

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement...

In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a contractive Banach function algebra which is not equivalent to a uniform algebra....

Recently in this Journal J. Esterlé gave a new proof of the Wiener Tauberian theorem for ${L}^{1}\left(\mathbf{R}\right)$ using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane. We here use essentially the same method to prove the analogous result for Beurling algebras ${L}_{\phi}^{1}\left(\mathbf{R}\right)$. Our estimates need a theorem of Hayman and Korenblum.

Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L¹(G) and the measure algebra M(G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C₀(Ω)” of the C*-algebra C₀(Ω) for a locally compact space Ω, recognizing this space as C(Ω̃), where Ω̃ is the hyper-Stonean envelope of Ω. We shall study the C*-algebra ${B}^{b}\left(\Omega \right)$ of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety of subsets...

The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces ${L}^{r}\left(\Omega \right)$ are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show that the standard...

A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.

We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A={\ell}^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras ${\ell}^{p}\left(\omega \right)$.

We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form ${L}^{r}\left(\Omega \right)$, and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.

We consider Fréchet algebras which are subalgebras of the algebra 𝔉 = ℂ [[X]] of formal power series in one variable and of 𝔉ₙ = ℂ [[X₁,..., Xₙ]] of formal power series in n variables, where n ∈ ℕ. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F)-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity of characters...

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